we present an alternative approach to the theory of free gibbs laws with convex potentials developed by dabrowski, guionnet, and shlyakhtenko. instead of solving sde's, we combine pde techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $m_n(\mathbb{c})_{sa}^m$ to prove the following. suppose $\mu_n$ is a probability measure on on $m_n(\mathbb{c})_{sa}^m$ given by uniformly convex and semi-concave potentials $v_n$, and suppose that the sequence $dv_n$ is asymptotically approximable by trace polynomials in a certain sense. then the moments of $\mu_n$ converge to a non-commutative law $\lambda$. moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$ agree and equal the limit of the normalized classical entropies of $\mu_n$. an upcoming paper will use the same techniques to obtain transport maps from $\lambda$ to a free semicircular family as the limit of transport maps for the matrix models $\mu_n$.